The question is from Permutation and Combination. A question which combines probability with progression. Our task is to identify the infinite series which is ultimately the probability of that given event. This section hosts a number of questions which are on par with CAT questions in difficulty on CAT Permutation and Combination, and CAT Probability.

Question 14: A and B take part in a duel. A can strike with an accuracy of 0.6. B can strike with an accuracy of 0.8. A has the first shot, post which they strike alternately. What is the probability that A wins the duel?

- \\frac{7}{10}\\)
- \\frac{15}{23}\\)
- \\frac{2}{3}\\)
- \\frac{11}{17}\\)

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A can win in the following scenarios:

A strikes in first shot.

A misses in the first shot, B misses in the second, A strikes in the third.

A misses in the 1st shot, B misses in the 2nd, A misses the 3rd, B misses the 4th and A strikes the 5th.

*And so on…*

P(A in 1^{st} shot) = 0.6

P(A in 3^{rd} shot) = 0.4 * 0.2 * 0.6 {A misses, then B misses and then A strikes}

P(A in 5^{st} shot) = 0.4 * 0.2 * 0.4 * 0.2 * 0.6 {A misses, then B misses and then A misses, then B misses, then A strikes}

Overall probability = Sum of all these

= 0.6 + 0.4 * 0.2 * 0.6 + 0.4 * 0.2 * 0.4 * 0.2 * 0.6 + …

Which is nothing but

0.6 + (0.4 * 0.2) * 0.6 + (0.4 * 0.2)^{2} * 0.6 + (0.4 * 0.2)^{3} * 0.6…

This is an infinite geometric progression with first term 0.6 and common ratio 0.4 * 0.2

Required Probability = \\frac{a}{1 - r}\\) = \\frac{0.6}{1 - 0.08}\\) = \\frac{0.6}{0.92}\\) = \\frac{30}{46}\\) = \\frac{15}{23}\\)

The question is **"What is the probability that A wins the duel?"**

Choice B is the correct answer.

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